Abstract
In this paper, we introduce and analyze a multistep Mann-type extragradient iterative algorithm by combining Korpelevich’s extragradient method, viscosity approximation method, hybrid steepest-descent method, Mann’s iteration method, and the projection method. It is proven that under appropriate assumptions, the proposed algorithm converges strongly to a common element of the fixed point set of infinitely many nonexpansive mappings and a strict pseudocontraction, the solution set of finitely many generalized mixed equilibrium problems (GMEPs), the solution set of finitely many variational inclusions and the solution set of a variational inequality problem (VIP), which is just a unique solution of a system of hierarchical variational inequalities (SHVI) in a real Hilbert space. The results obtained in this paper improve and extend the corresponding results announced by many others.
MSC:49J30, 47H09, 47J20, 49M05.
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1 Introduction
Let C be a nonempty closed convex subset of a real Hilbert space H and be the metric projection of H onto C. Let be a nonlinear mapping on C. We denote by the set of fixed points of S and by R the set of all real numbers. Let be a nonlinear mapping on C. We consider the following variational inequality problem (VIP): find a point such that
The solution set of VIP (1.1) is denoted by .
The VIP (1.1) was first discussed by Lions [1]. There are many applications of VIP (1.1) in various fields; see, e.g., [2–5]. It is well known that, if A is a strongly monotone and Lipschitz-continuous mapping on C, then VIP (1.1) has a unique solution. In 1976, Korpelevich [6] proposed an iterative algorithm for solving the VIP (1.1) in Euclidean space :
with a given number, which is known as the extragradient method. The literature on the VIP is vast and Korpelevich’s extragradient method has received great attention given by many authors, who improved it in various ways; see, e.g., [7–16] and references therein, to name but a few.
Let be a real-valued function, be a nonlinear mapping and be a bifunction. In 2008, Peng and Yao [9] introduced the generalized mixed equilibrium problem (GMEP) of finding such that
We denote the set of solutions of GMEP (1.2) by . The GMEP (1.2) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, Nash equilibrium problems in noncooperative games and others. The GMEP is further considered and studied; see, e.g., [7, 10, 14, 16–19]. In particular, if , then GMEP (1.2) reduces to the generalized equilibrium problem (GEP) which is to find such that
It was introduced and studied by Takahashi and Takahashi [20]. The set of solutions of GEP is denoted by .
If , then GMEP (1.2) reduces to the mixed equilibrium problem (MEP) which is to find such that
It was considered and studied in [21]. The set of solutions of MEP is denoted by .
If , , then GMEP (1.2) reduces to the equilibrium problem (EP) which is to find such that
It was considered and studied in [22–24]. The set of solutions of EP is denoted by .
On the other hand, let B be a single-valued mapping of C into H and R be a multivalued mapping with . Consider the following variational inclusion: find a point such that
We denote by the solution set of the variational inclusion (1.3). In particular, if , then . If , then problem (1.3) becomes the inclusion problem introduced by Rockafellar [25]. It is well known that problem (1.3) provides a convenient framework for the unified study of optimal solutions in many optimization related areas including mathematical programming, complementarity problems, variational inequalities, optimal control, mathematical economics, equilibria and game theory, etc. Let a set-valued mapping be maximal monotone. We define the resolvent operator associated with R and λ as follows:
where λ is a positive number.
In 1998, Huang [26] studied problem (1.3) in the case where R is maximal monotone and B is strongly monotone and Lipschitz continuous with . Subsequently, Zeng et al. [27] further studied problem (1.3) in the case which is more general than Huang’s one [26]. Moreover, the authors [27] obtained the same strong convergence conclusion as in Huang’s result [26]. In addition, the authors also gave the geometric convergence rate estimate for approximate solutions. Also, various types of iterative algorithms for solving variational inclusions have been further studied and developed; for more details, refer to [11, 19, 28–30] and the references therein.
Let S and T be two nonexpansive mappings. In 2009, Yao et al. [31] considered the following hierarchical variational inequality problem (HVIP): find hierarchically a fixed point of T, which is a solution to the VIP for monotone mapping ; namely, find such that
The solution set of the hierarchical VIP (1.4) is denoted by Λ. It is not hard to check that solving the hierarchical VIP (1.4) is equivalent to the fixed point problem of the composite mapping , i.e., find such that . The authors [31] introduced and analyzed the following iterative algorithm for solving the HVIP (1.4):
It is proved [[31], Theorem 3.2] that converges strongly to , which solves the hierarchical VIP:
Very recently, Kong et al. [7] introduced and considered the following system of hierarchical variational inequalities (SHVI) (over the fixed point set of a strictly pseudocontractive mapping) with a variational inequality constraint:
to find such that
In particular, if and , where is -strictly pseudocontractive for , SHVI (1.6) reduces to the following:
to find such that
The authors in [7] proposed the following algorithm for solving SHVI (1.6) and presented its convergence analysis:
where for all . In particular, if , then (1.8) reduces to the following iterative scheme:
In this paper, we introduce and study the following system of hierarchical variational inequalities (SHVI) (over the fixed point set of an infinite family of nonexpansive mappings and a strictly pseudocontractive mapping) with constraints of finitely many GMEPs, finitely many variational inclusions and the VIP (1.1):
Let M, N be two positive integers. Assume that
-
(i)
is a monotone and L-Lipschitzian mapping and is κ-Lipschitzian and η-strongly monotone with positive constants such that and where ;
-
(ii)
is a bifunction from to R satisfying (A1)-(A4) and is a proper lower semicontinuous and convex function with restriction (B1) or (B2), where ;
-
(iii)
is a maximal monotone mapping, and and are -inverse-strongly monotone and -inverse strongly monotone, respectively, where and ;
-
(iv)
is a sequence of nonexpansive self-mappings on C, is a ξ-strict pseudocontraction, is a nonexpansive mapping and is a ρ-contraction with coefficient ;
-
(v)
.
Then the objective is to find such that
In particular, whenever , the objective is to find such that
Motivated and inspired by the above facts, we introduce and analyze a multistep Mann-type extragradient iterative algorithm by combining Korpelevich’s extragradient method, viscosity approximation method, hybrid steepest-descent method, Mann’s iteration method and projection method. It is proven that under mild conditions, the proposed algorithm converges strongly to a common element of the solution set of finitely many GMEPs, the solution set of finitely many variational inclusions, the solution set of VIP (1.1) and the fixed point set of an infinite family of nonexpansive mappings and a strict pseudocontraction T, which is just a unique solution of the SHVI (1.10). The results obtained in this paper improve and extend the corresponding results announced by many others. For recent related work, we refer to [32] and the references therein.
2 Preliminaries
Throughout this paper, we assume that H is a real Hilbert space whose inner product and norm are denoted by and , respectively.
Then we have the following inequality:
We write to indicate that the sequence converges weakly to x and to indicate that the sequence converges strongly to x. Moreover, we use to denote the weak ω-limit set of the sequence , i.e.,
Lemma 2.1 Let H be a real Hilbert space. Then the following hold:
-
(a)
for all ;
-
(b)
for all and with ;
-
(c)
If is a sequence in H such that , it follows that
2.1 Nonexpansive type mappings
Let C be a nonempty closed convex subset of H. The metric (or nearest point) projection from H onto C is the mapping which assigns to each point the unique point satisfying the property
The following properties of projections are useful and pertinent to our purpose.
Proposition 2.1 Given any and . Then we have the following:
-
(i)
, ;
-
(ii)
, ;
-
(iii)
, .
Definition 2.1 A mapping is said to be
-
(a)
nonexpansive if
-
(b)
firmly nonexpansive if is nonexpansive, or equivalently, if T is 1-inverse-strongly monotone (1-ism),
Alternatively, T is firmly nonexpansive if and only if T can be expressed as , where is nonexpansive and I is the identity mapping on H. Note projections are firmly nonexpansive.
Definition 2.2 A mapping is said to be an averaged mapping if it can be written as the average of the identity I and a nonexpansive mapping, that is, , where and is nonexpansive.
Proposition 2.2 ([33])
Let be a given mapping. Then:
-
(i)
T is nonexpansive if and only if the complement is -ism.
-
(ii)
If T is ν-ism, then for , γT is -ism.
-
(iii)
T is averaged if and only if the complement is ν-ism for some . Indeed, for , T is α-averaged if and only if is -ism.
Let be given operators.
-
(i)
If for some and if S is averaged and V is nonexpansive, then T is averaged.
-
(ii)
T is firmly nonexpansive if and only if the complement is firmly nonexpansive.
-
(iii)
If for some and if S is firmly nonexpansive and V is nonexpansive, then T is averaged.
-
(iv)
The composite of finitely many averaged mappings is averaged. That is, if each of the mappings is averaged, then so is the composite . In particular, if is -averaged and is -averaged, where , then the composite is α-averaged, where .
-
(v)
If the mappings are averaged and have a common fixed point, then
We need some facts and tools which are listed as lemmas below.
Lemma 2.2 ([[35], Demiclosedness principle])
Let C be a nonempty closed convex subset of a real Hilbert space H. Let S be a nonexpansive self-mapping on C. Then is demiclosed at 0.
Let be an infinite family of nonexpansive self-mappings on C and be a sequence in . For any , define a mapping on C as follows:
Such a mapping is called the W-mapping generated by and .
Lemma 2.3 ([36])
Let C be a nonempty closed convex subset of a real Hilbert space H. Let be a sequence of nonexpansive self-mappings on C such that and let be a sequence in for some . Then, for every and the limit exists, where is defined as in (2.1).
Remark 2.1 ([[37], Remark 3.1])
It can be known from Lemma 2.4 that if D is a nonempty bounded subset of C, then for there exists such that for all .
Remark 2.2 ([[37], Remark 3.2])
Utilizing Lemma 2.4, we define a mapping as follows:
Such a W is called the W-mapping generated by and . Since is nonexpansive, is also nonexpansive. For a bounded sequence in C, we put . Hence, it is clear from Remark 2.1 that for an arbitrary , there exists such that for all
This implies that .
Lemma 2.4 ([36])
Let C be a nonempty closed convex subset of a real Hilbert space H. Let be a sequence of nonexpansive self-mappings on C such that , and let be a sequence in for some . Then .
It is clear that, in a real Hilbert space H, is ξ-strictly pseudocontractive if and only if the following inequality holds:
This immediately implies that if T is a ξ-strictly pseudocontractive mapping, then is -inverse-strongly monotone; for further detail, we refer to [38] and the references therein. It is well known that the class of strict pseudocontractions strictly includes the class of nonexpansive mappings and that the class of pseudocontractions strictly includes the class of strict pseudocontractions. In addition, for the extension of strict pseudocontractions, please the reader refer to [39] and the references therein.
Proposition 2.4 ([[38], Proposition 2.1])
Let C be a nonempty closed convex subset of a real Hilbert space H and be a mapping.
-
(i)
If T is a ξ-strictly pseudocontractive mapping, then T satisfies the Lipschitzian condition
-
(ii)
If T is a ξ-strictly pseudocontractive mapping, then the mapping is demiclosed at 0, that is, if is a sequence in C such that and , then .
-
(iii)
If T is ξ-(quasi-)strict pseudocontraction, then the fixed point set of T is closed and convex so that the projection is well defined.
Proposition 2.5 ([40])
Let C be a nonempty closed convex subset of a real Hilbert space H. Let be a ξ-strictly pseudocontractive mapping. Let γ and δ be two nonnegative real numbers such that . Then
2.2 Mixed equilibrium problems
We list some elementary conclusions for the MEP.
It was assumed in [9] that is a bifunction satisfying conditions (A1)-(A4) and is a lower semicontinuous and convex function with restriction (B1) or (B2), where
(A1) for all ;
(A2) Θ is monotone, i.e., for any ;
(A3) Θ is upper-hemicontinuous, i.e., for each ,
(A4) is convex and lower semicontinuous for each ;
(B1) for each and , there exists a bounded subset and such that for any ,
(B2) C is a bounded set.
Proposition 2.6 ([21])
Assume that satisfies (A1)-(A4) and let be a proper lower semicontinuous and convex function. Assume that either (B1) or (B2) holds. For and , define a mapping as follows:
for all . Then the following hold:
-
(i)
for each , is nonempty and single-valued;
-
(ii)
is firmly nonexpansive, that is, for any ,
-
(iii)
;
-
(iv)
is closed and convex;
-
(v)
for all and .
2.3 Monotone operators
Definition 2.3 Let T be a nonlinear operator with the domain and the range . Then T is said to be
-
(i)
monotone if
-
(ii)
β-strongly monotone if there exists a constant such that
-
(iii)
ν-inverse-strongly monotone if there exists a constant such that
It can easily be seen that if T is nonexpansive, then is monotone. It is also easy to see that the projection is 1-ism. Inverse strongly monotone (also referred to as co-coercive) operators have been applied widely in solving practical problems in various fields, for instance, in traffic assignment problems; see [41, 42]. On the other hand, it is obvious that if A is ζ-inverse-strongly monotone, then A is monotone and -Lipschitz continuous. Moreover, we also have, for all and ,
So, if , then is a nonexpansive mapping from C to H.
Let C be a nonempty closed convex subset of a real Hilbert space H. We introduce some notations. Let λ be a number in and let . Associated with a nonexpansive mapping , we define the mapping by
where is an operator such that, for some positive constants , F is κ-Lipschitzian and η-strongly monotone on C; that is, F satisfies the conditions:
Lemma 2.5 (see [[43], Lemma 3.1])
is a contraction provided ; that is,
where .
Remark 2.3 (i) Since F is κ-Lipschitzian and η-strongly monotone on C, we get . Hence, whenever , we have . So, .
(ii) In Lemma 2.5, put and . Then we know that , and
Lemma 2.6 Let be a monotone mapping. The characterization of the projection (see Proposition 2.1(i)) implies
Finally, recall that a set-valued mapping is called monotone if for all , and imply
A set-valued mapping is called maximal monotone if is monotone and for each . We denote by the graph of . It is well known that a monotone mapping is maximal if and only if, for , for every implies . Next we provide an example to illustrate the concept of maximal monotone mapping.
Let be a monotone, k-Lipschitz-continuous mapping and let be the normal cone to C at , i.e.,
Define
Then is maximal monotone (see [25]) such that
Let be a maximal monotone mapping. Let be two positive numbers.
Lemma 2.7 (see [44])
We have the resolvent identity
In terms of Huang [26] (see also [27]), we have the following property for the resolvent operator .
Lemma 2.8 is single-valued and firmly nonexpansive, i.e.,
Consequently, is nonexpansive and monotone.
Lemma 2.9 ([11])
Let R be a maximal monotone mapping with . Then for any given , is a solution of problem (1.3) if and only if satisfies
Lemma 2.10 ([27])
Let R be a maximal monotone mapping with and let be a strongly monotone, continuous and single-valued mapping. Then for each , the equation has a unique solution for .
Lemma 2.11 ([11])
Let R be a maximal monotone mapping with and be a monotone, continuous and single-valued mapping. Then for each . In this case, is maximal monotone.
2.4 Technical lemmas
The following lemma plays a key role in proving strong convergence of the sequences generated by our algorithms.
Lemma 2.12 ([45])
Let be a sequence of nonnegative real numbers satisfying the property:
where and are such that
-
(i)
;
-
(ii)
either or ;
-
(iii)
where , for all .
Then .
Lemma 2.13 ([46])
Let and be the sequences of nonnegative real numbers and a sequence of real numbers, respectively, such that and . Then .
3 Main results
In this section, we will introduce and analyze a multistep Mann-type extragradient iterative algorithm for finding a solution of SHVI (1.10) (over the fixed point set of an infinite family of nonexpansive mappings and a strict pseudocontraction) with constraints of several problems: finitely many GMEPs, finitely many variational inclusions and VIP (1.1) in a real Hilbert space. This algorithm is based on Korpelevich’s extragradient method, viscosity approximation method, hybrid steepest-descent method, Mann’s iteration method and projection method. We prove the strong convergence of the proposed algorithm to a unique solution of SHVI (1.10) under suitable conditions.
We are now in a position to state and prove the main result in this paper.
Theorem 3.1 Let C be a nonempty closed convex subset of a real Hilbert space H. Let M, N be two positive integers. Let be a bifunction from to R satisfying (A1)-(A4) and be a proper lower semicontinuous and convex function with restriction (B1) or (B2), where . Let be a maximal monotone mapping and let and be -inverse-strongly monotone and -inverse-strongly monotone, respectively, where , . Let be a sequence of nonexpansive self-mappings on C and be a sequence in for some . Let be a ξ-strictly pseudocontractive mapping, be a nonexpansive mapping and be a ρ-contraction with coefficient . Let be a -inverse-strongly monotone mapping, and be κ-Lipschitzian and η-strongly monotone with positive constants such that and where . Assume that SHVI (1.10) over has a solution. Let , , , and , where and . For arbitrarily given , let be a sequence generated by
where for all . Suppose that
(C1) ;
(C2) ;
(C3) and for all ;
(C4) and ;
(C5) ;
(C6) and .
If is bounded, then converges strongly to a unique solution of SHVI (1.10) provided .
Proof For , put
for all and
for all , and . Then we have and . In addition, in terms of conditions (C1), (C2), and (C4), without loss of generality, we may assume that for some , for some , and for all .
One can readily see that are nonexpansive for all ; see [7] (also [47]).
Next, we divide the remainder of the proof into several steps.
Step 1. is bounded.
Take a fixed arbitrarily. Utilizing (2.2) and Proposition 2.6(ii) we have
Utilizing (2.2) and Lemma 2.8 we have
which together with the last inequality, implies that
Note that for all and for . Hence, from (3.1) and (3.4), it follows that
Put for each . Then, by Proposition 2.1(ii), we have
Further, by Proposition 2.1(i), we have
So, we obtain from (3.6)
Since for all , utilizing Proposition 2.5 and Lemma 2.1(b), from (3.5) and (3.8), we conclude that
Noticing the boundedness of , we get for some . Moreover, utilizing Lemma 2.5 we have from (3.1)
By induction, we can derive
Consequently, is bounded (due to ) and so are the sequences , , , , and .
Step 2. , , , , and as .
From (3.1) and (3.9), it follows that
where . This together with and implies that
Note that . Hence, taking into account the boundedness of and , we deduce from (3.12) that
Furthermore, for simplicity, we write for all . Then we have
which immediately yields
So, utilizing Proposition 2.1(i) we get
Since and (due to (C6)), we know from the boundedness of , , and that
Taking into account that , we obtain from (3.13) and (3.14)
Next let us show that . As a matter of fact, from (3.4) and (3.8) it follows that
Utilizing Lemma 2.1(b), from (3.9) and (3.16), we obtain
which immediately implies that
Since , (due to (3.14)) and , , are bounded, we get
In the meantime, from (3.8) and (3.9) it is not hard to find that
Now, let us show that . In fact, observe that
and
for and . Combining (3.18), (3.19), and (3.20), we get
which hence implies that
Since , , and where and , we deduce from (3.14) and the boundedness of , , that
where and .
Furthermore, by Proposition 2.6(ii) and Lemma 2.1(a) we have
which implies that
By Lemma 2.1(a) and Lemma 2.8, we obtain
which immediately leads to
Combining (3.18) and (3.23) we conclude that
which yields
Since , and where , we deduce from (3.21) and the boundedness of , , , that
Also, combining (3.3), (3.18), and (3.22) we deduce that
which leads to
Since , and where , we deduce from (3.21) and the boundedness of , , that
Hence from (3.24) and (3.25) we get
and
respectively. Thus, from (3.26) and (3.27) we obtain
In addition, it is clear that
Thus, we conclude from Remark 2.2, (3.15), and the boundedness of that
Noting that , we have from (3.13) and (3.28) that
Again, noting that , we obtain from (3.17) and (3.30)
Furthermore, from (3.1) we find that and hence
which immediately leads to
Consequently, from (3.14), (3.31), and we get
Step 3. .
Since H is reflexive and is bounded, there exists at least a weak convergence subsequence of . Hence it is well known that . Now, take an arbitrary . Then there exists a subsequence of such that . From (3.24)-(3.26), (3.28), (3.30), and (3.31), we have , , , , and , where and . Utilizing Proposition 2.4(ii) and Lemma 2.2, we deduce from (3.29) and (3.32) that and (due to Lemma 2.4). Next, we prove that . As a matter of fact, since is -inverse-strongly monotone, is a monotone and Lipschitz-continuous mapping. It follows from Lemma 2.11 that is maximal monotone. Let , i.e., . Again, since , , , we have
that is,
In terms of the monotonicity of , we get
and hence
In particular,
Since (due to (3.24)) and (due to the Lipschitz continuity of ), we conclude from and that
It follows from the maximal monotonicity of that , i.e., . Therefore, . Next we prove that . Since , , , we have
By (A2), we have
Let for all and . This implies that . Then we have
By (3.25), we have as . Furthermore, by the monotonicity of , we obtain . Then by (A4) we obtain
Utilizing (A1), (A4), and (3.34), we obtain
and hence
Letting , we have, for each ,
This implies that and hence . Furthermore, let us show that . In fact, define
where . Then is maximal monotone and if and only if ; see [25]. Let . Then we have , and hence . So, we have for all . On the other hand, from and , we get , and hence,
Therefore, from and , we have
Hence, it is easy to see that as . Since is maximal monotone, we have , and hence, . Consequently, . This shows that .
Step 4. converges strongly to a unique solution of SHVI (1.10).
Indeed, according to , we can take a subsequence of satisfying
Without loss of generality, we may further assume that ; then due to Step 3. Since is a solution of SHVI (1.10), we get
Repeating the argument of (3.35), we have
From (3.1) and (3.9), it follows that (noticing that and )
where . It turns out that
where , and
In terms of conditions (C5) and (C6), we conclude from that and . Note that and , where . Consequently, utilizing Lemma 2.13 we obtain
So, applying Lemma 2.12 to (3.37), we infer that . The proof is complete. □
Remark 3.1 In Theorem 3.1, whenever , the iterative scheme (3.1) reduces to the following one:
where for all . Assume that the SHVI (1.11) has a solution and that all the conditions in Theorem 3.1 are satisfied. If is bounded, then converges strongly to a unique solution of SHVI (1.11) provided .
Next we consider a special case of SHVI (1.10). In SHVI (1.10), put , and . In this case, the objective is to find such that
Utilizing Theorem 3.1 we immediately derive the following.
Corollary 3.1 Let C be a nonempty closed convex subset of a real Hilbert space H. Let M, N be two positive integers. Let be a bifunction from to R satisfying (A1)-(A4) and be a proper lower semicontinuous and convex function with restriction (B1) or (B2), where . Let be a maximal monotone mapping and let and be -inverse-strongly monotone and -inverse-strongly monotone, respectively, where , . Let be a sequence of nonexpansive self-mappings on C and be a sequence in for some . Let be a ξ-strictly pseudocontractive mapping, be a nonexpansive mapping and be a ρ-contraction with coefficient . Let be a -inverse-strongly monotone mapping. Assume that the SHVI (3.39) has a solution, where . Let , , and , where and . For arbitrarily given , let be a sequence generated by
where for all . Suppose that
(C1) ;
(C2) ;
(C3) and for all ;
(C4) and ;
(C5) ;
(C6) and .
If is bounded, then converges strongly to a unique solution of the SHVI (3.39) provided .
In Theorem 3.1, putting and , we obtain the following.
Corollary 3.2 Let C be a nonempty closed convex subset of a real Hilbert space H. Let be a bifunction from to R satisfying (A1)-(A4), be a proper lower semicontinuous and convex function with restriction (B1) or (B2), and be -inverse strongly monotone. Let be a maximal monotone mapping and be -inverse-strongly monotone, for . Let be a sequence of nonexpansive self-mappings on C and be a sequence in for some . Let be a ξ-strictly pseudocontractive mapping, be a nonexpansive mapping and be a ρ-contraction with coefficient . Let be a -inverse-strongly monotone mapping, and be κ-Lipschitzian and η-strongly monotone with positive constants such that and where . Assume that the SHVI (1.10) has a solution, where . Let , , , and for . For arbitrarily given , let be a sequence generated by
where for all . Suppose that
(C1) ;
(C2) ;
(C3) and for all ;
(C4) and ;
(C5) ;
(C6) and .
If is bounded, then converges strongly to a unique solution of the SHVI (1.10) provided .
In Theorem 3.1, putting , we obtain the following.
Corollary 3.3 Let C be a nonempty closed convex subset of a real Hilbert space H. Let be a bifunction from to R satisfying (A1)-(A4), be a proper lower semicontinuous and convex function with restriction (B1) or (B2), and be -inverse strongly monotone. Let be a maximal monotone mapping and be -inverse-strongly monotone. Let be a sequence of nonexpansive self-mappings on C and be a sequence in for some . Let be a ξ-strictly pseudocontractive mapping, be a nonexpansive mapping and be a ρ-contraction with coefficient . Let be a -inverse-strongly monotone mapping, and be κ-Lipschitzian and η-strongly monotone with positive constants such that and where . Assume that the SHVI (1.10) has a solution, where . Let , , , and . For arbitrarily given , let be a sequence generated by
where for all . Suppose that
(C1) ;
(C2) ;
(C3) and for all ;
(C4) and ;
(C5) ;
(C6) and .
If is bounded, then converges strongly to a unique solution of the SHVI (1.10) provided .
Remark 3.2 It is obvious that our iterative scheme (3.1) is very different from Yao et al.’s iterative one (1.5) and Kong et al.’s iterative one (1.8). Here, the three-step iterative scheme in [[7], Algorithm I] is extended to develop our five-step iterative scheme (3.1) for the SHVI (1.10) by combining Korpelevich’s extragradient method, viscosity approximation method, hybrid steepest-descent method [48], Mann’s iteration method and projection method. It is worth pointing out that under the lack of the assumptions similar to those in [[31], Theorem 3.2], e.g., is bounded and , the sequence generated by (3.1) converges strongly to a point , which is a unique solution of the SHVI (1.10) (over the fixed point set of an infinite family of nonexpansive mappings and a ξ-strictly pseudocontractive mapping T). It is worth emphasizing that the nonexpansive mapping T in (1.5) is extended to a ξ-strictly pseudocontractive mapping T in (3.1) and the VIP in SHVI (1.6) is extended to the setting of finitely many GMEPs and finitely many variational inclusions in SHVI (1.10).
Remark 3.3 Our Theorem 3.1 improves, extends, supplements and develops Yao et al. [[31], Theorems 3.1 and 3.2] and Kong et al. [[7], Theorem 17] in the following aspects:
-
(a)
Our SHVI (1.10) with the unique solution satisfying
is more general than the problem of finding a point satisfying in [31] and than the problem of finding a point satisfying in [[7], Theorem 17]. It is worth emphasizing that S is nonexpansive if and only if the complement is -inverse-strongly monotone; see [19].
-
(b)
Our five-step iterative scheme (3.1) for SHVI (1.10) is more flexible, more advantageous and more subtle than Kong et al.’s three-step iterative one in [[7], Algorithm I] and than Yao et al.’s two-step iterative one (1.5) because it can be used to solve several kinds of problems, e.g., the SHVI, the HVIP and the problem of finding a common point of four sets: , , and . In addition, our Theorem 3.1 drops the crucial requirements in [[31], Theorem 3.2] that , , and is bounded, generalizes [[7], Theorem 17] from one nonlinear mapping T to an infinite family of nonlinear mappings and T and extends [[7], Theorem 17] to the setting of finitely many GMEPs and finitely many variational inclusions.
-
(c)
The argument techniques in our Theorem 3.1 are very different from the argument ones in [[31], Theorems 3.1 and 3.2] and from the argument ones in [[7], Theorem 17] because we make use of the properties of the W-mappings (see Lemmas 2.4 and 2.5), the properties of resolvent operators and maximal monotone mappings (see Proposition 2.4 and Lemmas 2.9-2.13), the inclusion problem ( for maximal monotone operator ) (see (2.3)) and the contractive coefficient estimates for the contractions associating with nonexpansive mappings (see Lemma 2.7).
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Acknowledgements
Lu-Chuan Ceng was partially supported by the National Science Foundation of China (11071169), Innovation Program of Shanghai Municipal Education Commission and PhD Program Foundation of Ministry of Education of China (20123127110002). Jen-Chih Yao was partially supported by the grant NSC 102-2115-M-037-002-MY3.
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Ceng, LC., Sahu, D.R. & Yao, JC. A unified extragradient method for systems of hierarchical variational inequalities in a Hilbert space. J Inequal Appl 2014, 460 (2014). https://doi.org/10.1186/1029-242X-2014-460
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DOI: https://doi.org/10.1186/1029-242X-2014-460